How can I solidify without ruining spiked edges?
I would Solidify the outer side, and adjust the spike manually.  


Use the Solidify Modifier.
Set Offset to 1 and adjust Thickness to your liking.  





You can adjust Inner/Outer Crease, if you're also using the Subdivision modifier.
The Subdivision modifier should be AFTER the Solidify Modifier.




Hit the Apply button to apply the Solidify Modifier.


Go to Edit mode
Select the new vertices and scale it down to make it pointy using the Scale Tool.
Shortcut: S